One property of a planar curve whose convex hull covers a given convex figure

TL;DR

This paper proves a conjecture relating the length of a planar curve to the perimeter and diameter of a convex figure it covers, establishing a lower bound and analyzing equality cases.

Contribution

It confirms a conjecture connecting curve length and convex hull properties, providing a new inequality and characterizing equality scenarios.

Findings

Established a lower bound for the length of a curve covering a convex figure.

Proved the conjecture by Akopyan and Vysotsky.

Analyzed cases where the inequality becomes equality.

Abstract

In this note, we prove the following conjecture by A. Akopyan and V. Vysotsky: If the convex hull of a planar curve $\gamma$ covers a planar convex figure $K$, then $\operatorname{length}(\gamma) \geq \operatorname{per} (K) - \operatorname{diam} (K)$. In addition, all cases of equality in this inequality are studied.